Lecture held before norges tekniske hogskole, Norway

S h i p m o t i o n s P r o f . i r . J - G e r r i t s m a R e p o r t n r : 150-P L e c t u r e h e l d b e f o r e N o r g e s T e k n i s k e H ^ g s k o l e , I n s t i t u t t f o r S k p i s b y g g i n g , T r o n d h e i m , Norway, 14-15 S e p t e m b e r '64 D e l f t U n i v e r s i t y of T e c h n o l o g y

Ship Hydromechanics Laboratory Mekelweg 2

2628 CD D E L F T The Netherlands Phone 015 -786882

Meddelelse SKB I l / M l SHIP MOTIONS Guest Lectures by P r o f . i r , J , Gerritsma Trondheim September 14.- 15. 1964

N O R G E S T E K N I S K E H 0 G S K O L E

I N S T I T U T T F O R S K I P S B Y G G I N G I i T R O N D H E I M — N . T . H .

I n t r o d u c t i o n o

I n t h i s l e c t u r e I s h a l l t r e a t f o r you some a s p e c t s o f t h e

p r o b l e m o f s h i p m o t i o n s i n s t i l l w a t e r . The s o l u t i o n o f t h i s p r o b l e m i s i m p o r t a n t f o r t h e c a l c u l a t i o n and a n a l y s i s o f shipir.otions and r e -l a t e d phenomena such as b e n d i n g moments, s-lamming, t h e s h i p p i n g o f green w a t e r and B O on, i n r e g u l a r and i r r e g u l a r seas.

To show you t h e i m p o r t a n c e o f t h e o s c i l l a t i o n s i n s t i l l w a t e r we s h a l l s t u d y f i r s t o f a l l , i n a v e r y s i m p l i f i e d way, t h e m o t i o n s o f a S h i p i n l o n g r e g u l a r and l o n g i t u d i n a l head waves»

The s h i p has no f o r w a r d speed and t h o c r o s s s e c t i o n s a r e w a l l

s i d e d i n t h e v i c i n i t y o f t h e l o a d v / a t e r l i n e . Consider a r i g h t hand c o o r d i n a t e s y s t e m , f i x e d in space, ^'^^^^q'^q^' ^ second system i s

a t t a c h e d t o t h e s h i p : ( x y z ) . The o r i g i n o f t h i s system l i e s i n t h e c e n t r e o f g r a v i t y o f t h e s h i p ( G ) . For t h e s h i p a t r e s t t h e two co-o r d i n a t e systems c co-o i n c i d e .

The s u r g i n g m o t i o n i s n e g l e c t e d and t h e heave and p i t c h o f t h e s h i p a r e d e s c r i b e d by z^Ct) and 0( t ) a

I 2 -A l o n g r e g u l a r wave ^ , v / i t h s m a l l a m p l i t u d e r r u n s i n t h e d i r e c t i o n o f t h e n e g a t i v e a x i s . The s u r f a c e e l e v a t i o n o f t h i s wave i s g i v e n by: ^ = r cos ( k x^ + C*} t ) ,

where k = i s t h e wave number and ^ = 7 ^ i s t h e c i r c u l a r f r e -quency o f t h e wave»

Because we n e g l e c t t h e s u r g i n g m o t i o n and by assumin;-^ t h a t the m o t i o n a m p l i t u d e s a r e v e r y s m a l l , we m.ay w r i t e : X ^ X o o The v e r t i c a l d i s p l a c e m e n t o f a s t r i p o f t h e s h i p a t a d i s t a n c e x from t h e c e n t r e o f g r a v i t y i s g i v e n b y : z - X 6 0 o The d i s p l a c e m e n t r e l a t i v e t o t h e water s u r f a c e i s : z - X e - f . o The r e l a t i v e v e r t i c a l v e l o c i t y i s g i v e n b y : ! The r e l a t i v e v e r t i c a l a c c e l e r a t i o n w i l l be: i z - X 8 - « The v e r t i c a l f o r c e s a c t i n g on Q s t r i p o f l e n g t h dx o f t h e s h i p a r e r e l a t e d by ,the f o l l o w i n g e q u a t i o n : (Newton) ( — d x ) ( z - x 8 ) = / ) g A dx - 2 / 0 e Y ( z ^ - X 0 - ^ ) dx - w dx g o / X / O r¬ N ' (z - X Ó - ^ ) dx + m * Cz^ - X Ö ) d x , where: w' i s t h e w e i g h t o f t h e s h i p p e r u n i t l e n g t h ; A i s t h e c r o s s s e c t i o n under t h e l o a d w a t e r l i n e ^ X Y i s t h e h a l f w i d t h o f t h e w a t e r l i n e a t x; N' i s a c o e f f i c i e n t ( t h e s e c t i o n a l damping c o e f f i c i e n t ) ; m' i s a c o e f f i c i e n t ( t h e s e c t i o n a l added mass).

Here we assume t h a t t h e p r e s s u r e i n t h e wave i s n o t d i s t u r b e d by t h e presence o f t h e s h i p .

The t o t a l v e r t i c a l f o r c e on t h e s h i p i s f o u n d by i n t e g r a t i o n . We have: pg

/ k_^dx = J

vi' dx, because a t r e s t t h e w e i g h t o f t h e s h i p equals t h e w e i g h t o f t h e d i s p l a c e m e n t . A l s o we have:

14

dx = o T t h e mass o f t h e d i s p l a c e m e n t , S ' 2 p g y Y dx = |OgA^j.= e t h e w a t e r l i n e a r e a t i m e s t h e s p e c i f i c g r a v i t y o f t h e w a t e r . T h i s c o e f f i c i e n t i s c a l l e d : t h e r e s t o r i n g f o r c e c o e f f i c i e n t ,

^ f> ë / Y x d x = + /Og S^^ = g t h e f i r s t moment o f the w a t e r p l a n e a r e a w i t h r e s p e c t t o a t r a n s v e r s e a x i s t h r o u g h G, assuming t h a t G l i e s i n the w a t e r p l a n e ; ƒ N' dx = h t h e damping c o e f f i c i e n t f o r heave; ƒ N' x d x = e a dynamic c r o s s c o u p l i n g c o e f f i c i e n t ; L

ƒ

• m dx = a t h e added mass; L ƒ m X dx = d a dynamic c r o s s c o u p l i n g c o e f f i c i e n t , L P e - a r r a n g i n g a f t e r i n t e g r a t i o n we f i n d t h e heave e q u a t i o n : (a + iOTDz + b z + C Z - ' d 0 - e Ó - g O = 2 g / Y i d x + ' o o o T + / N ' i dx +

ƒ

m'l^' i C c j t + 6„) dx = F e ^ ^ a By, t a k i n g t h e moment o f t h e f o r c e s on t h e s t r i p w i t h r e s p e c t t o G, we f i n d i n a s i m i l a r way the p i t c h e q u a t i o n : j

4

-E z - G z = M e o o a

These e q u a t i o n s f o r heave and } ) i t c h a r e second o r d e r l i n e a r d i f -f e r e n t i a l e q u a t i o n s , w h i c h a r e c o u p l e d . I n t h e heave e q u a t i o n we liave terms which c o n t a i n t h e p i t c h angle " 0 " and i n t h e p i t c h moment equa-t i o n equa-t h e heave "z appears equa-t o be p r e s e n equa-t . The c o u p l i n g a r i s e s from the f a c t t h a t t h e d i s t r i b u t i o n o f t h e f o r c e s a l o n g t)ie h u l l i s n o t sym-m e t r i c w i t h r e s p e c t t o G. For i n s t a n c e t h e tersym-m g 0 e x i s t s f o r a f o r

and a f t non symmetric w a t e r l i n e , because g = g ^^j^i where 3^^^ ^ j , ^ ^ s t a t i c a l moment o f t h e w a t e r p l a n e w i t h r e s p e c t t o a t r a n s v e r s e a x i s

t h r o u g h G. For a s y m m e t r i c a l w a t e r l i n e g = 0 . For a s y m m e t r i c a l s h i p -form a t zero speed t h e o t h e r c o u p l i n g c o e f f i c i e n t s a r e a l s o e q u a l t o z e r o . I n t h a t case:

d = e = g = D = E •= G = 0 ,

I t has t o be emphasized t h a t up t i l l now our p r o b l e m i s v e r y s i m p l i -f i e d : t h e wave l e n g t h i s v e r y l a r g e i n c o m p a r i s o n w i t h t h e s h i p l e n g t h and c o n s e q u e n t l y t h e boyancy i n t h e wave i s t a k e n as i n t h e case o f a h y d r o s t a t i c c a l c u l a t i o n ; a l s o t h e e f f e c t o f f o r w a r d speed i s n o t c o n

-terms a r e s i t u a t e d i n t h e r i g h t hand s i d e s ^ These r i g h t hand s i d e s g i v e t h e f o r c e s and moments on a non h e a v i n g and non p i t c h i n g s h i p i n waves. vVe have t h e same s i t u a t i o n wlien a more r e f i n e d a n a l y s i s i s a p p l i e d t o t h e p r o b l e m , a t l e a s t as a f i r s t a p p r o x i m a t i o n .

s i d e r e d o

The l e f t hand s i d e s o f t h e e q u a t i o n s c o n t a i n o n l y q u a n t i t i e s w h i c h d e s c r i b e t h e m o t i o n i n s t i l l w a t e r . The wave e l e v a t i o n )

The c a l c u l a t i o n o f s h i p r . o t i o n s i n r e g u l a r head waves by u s i n g a s t r i p t h e o r y , has been d i s c u s s e d i n a number o f p a p e r s , Hecent c o n t r i b u t i o n s v/ero g i v e n by Korvin»Kroukovsky and Jacobs [ l ] , Fay [ 2 j , 7/atanabe [jJ. and Fukuda [k] .

I n these papers the i n f l u e n c e o f f o r w a r d speed on the h y d r o d y namic f o r c e s i s c o n s i d e r e d and dynamic c r o s s c o u p l i n g terms a r e i n -c l u d e d i n the e q u a t i o n s o f m o t i o n , w h i -c h are assumed t o d e s -c r i b e the h e a v i n g and p i t c h i n g m o t i o n s .

I n e a r l i e r work [ 5 ] i t was shown t h a t a r e l a t i v e l y s m a l l i n f l u e n -ce o f speed e x i s t s on t h e damping coe f f n . c i e n t s , the added mass and the e x c i t i n g f o r c e s , a t l e a s t f o r t h e case o f head waves and f o r speeds w h i c h are o f p r a c t i c a l i n t e r e s t . On the o t h e r hand, f o r w a r d speed has an i m p o r t a n t e f f e c t on some o f the dynamic c r o s s - c o u p l i n g c o e f f i c i e n t s . A l t h o u g h , a t a f i r s t g l a n c e these terms c o u l d be r e garded as second o r d e r q u a n t i t i e s , i t was p o i n t e d o u t by K o r v i n -K r o u k o v s k y [1] and a l s o by Fay [ s ] t h a t t h e y can be v e r y im^portant f o r the a m p l i t u d e s and phases o f t h e m o t i o n s . T h i s has been c o n f i r m e d i n [5]where t h e c o u p l i n g terms a r e n e g l e c t e d i n a c a l c u l a t i o n o f t h e h e a v i n g and p i t c h i n g m o t i o n s . I n t h i s c a l c u l a t i o n we used c o e f f i -c i e n t s o f t h e m o t i o n e q u a t i o n s , w h i -c h were d e t e r m i n e d by f o r -c e d osc i l l a t i o n t e s t s . I n oscomparison w i t h t h e osc a l osc u l a t i o n v/here t h e osc r o s s -c o u p l i n g terms are i n -c l u d e d and a l s o i n -comparison w i t h t h e measured m o t i o n s , an i m p o r t a n t i n f l u e n c e i s o b s e r v e d , as shown i n F i g u r e 1 , w h i c h i s t a k e n from r e f e r e n c e [ 5 j . F u r t h e r a n a l y s i s showed t h a t t h e d i s c r e p e n c i e s between the c o u p l e d and uncoupled m o t i o n s were m a i n l y due t o the damping c r o s s - c o u p l i n g t e r m s .

The i n f l u e n c e o f f o r w a r d speed has been d i s c u s s e d t o some ex-t e n ex-t i n Vossers's ex-t h e s i s [ 6 ] . From a f i r s t o r d e r s l e n d e r body t h e o r y i t was found t h a t the d i s t r i b u t i o n o f t h e hydrodynamic f o r c e s a l o n g an o s c i l l a t i n g s l e n d e r body i s n o t i n f l u e n c e d by f o r w a r d speed. VoBsers concluded t h a t the i n c l u s i o n o f speed dependent damping c r o s s - c o u p l i n g terms i s n o t i n agreement w i t h the use o f .a s t r i p t h e o r y . I n view o f t h e above m e n t i o n e d r e s u l t s such a s i m p l i f i c a -t i o n does n o -t h o l d f o r a c -t u a l s h i p f o r m s .

6

-For s y m m e t r i c a l s l i i p f o r n r s a t forv/ard speed, i t v/as shown by Timman and Newman ( 7 ]t h a t the damping c r o s s - c o u p l i n g c o e f f i c i e n t s f o r heave and p i t c h are e q u a l i n m a g n i t u d e , b u t o p p o s i t e i n s i g n . T h e i r c o n c l u s i o n i s v a l i d f o r t h i n or s l e n d e r submerged o f s u r f a c e s h i p s and a l s o f o r non s l e n d e r b o d i e s .

Golovato's work [8] and some o f our e x p e r i m e n t s [ 5 ] on o s c i l l a t i n g shipmodels c o n f i r m e d t h i s f a c t f o r a c t u a l s u r f a c e s l i i p s t o a c e r t a i n e x t e n t .

The e f f e c t s o f f o r w a r d speed are i n d e e d v e r y i r . p o r t a n t f o r the c a l c u l a t i o n o f s h i p m o t i o n s i n waves. The t w o - d i m e n s i o n a l s o l u t i o n s f o r damping and added mass o f o s c i l l a t i n g c y l i n d e r s on a f r e e s u r -f a c e , as g i v e n by G r i m [ 9 ] a n d T a s a i[ l o ] show a v e r y s a t i s f a c t o r y agree-ment w i t h e x p e r i m e n t a l r e s u l t s . When t h e e f f e c t s o f f o r w a r d speed can be e s t i m a t e d w i t h s u f f i c i e n t a c c u r a c y , such t w o - d i m e n s i o n a l v a l u e s may be used t o c a l c u l a t e t h o t o t a l hydrodynamic f o r c e s and moments on a s h i p , p r o v i d e d t h a t i n t e g r a t i o n over t h e s h i p l e n g t h i s p e r m i s s i b l e .

I n o r d e r t o s t u d y t h o speed e f f e c t on an o s c i l l a t i n g s h i p f o r m i n more d e t a i l , a s e r i e s o f f o r c e d o s c i l l a t i n g e x p e r i m e n t s was des i g n e d . The main o b j e c t o f thedese e x p e r i m e n t des wades t o f i n d t h e d i des t r i -b u t i o n o f t h e hydrodynamic f o r c e s a l o n g the l e n g t h o f t h e s h i p as a

f u n c t i o n o f f o r w a r d speed and f r e q u e n c y o f o s c i l l a t i o n . 2. The e x p e r i m e n t s .

The o s c i l l a t i o n t e s t s were c a r r i e d out w i t h a 2 . 3 meter model o f t h e S i x t y S e r i e s , h a v i n g a b l o c k c o e f f i c i e n t C^ = . 7 0 . The main dimensions a r e g i v e n i n Table 1 . The model i s made o f p o l y e s t e r , r e -i n f o r c e d w -i t h f -i b r e g l a s s , and c o n s -i s t s o f seven s e p a r a t e s e c t -i o n s o f e q u a l l e n g t h . Each o f the s e c t i o n s has two e n d - b u l k h e a d s . The w i d t h o f t h e gap between two s e c t i o n s i s one m i l l i m e t e r . The

sec-t i o n s are n o sec-t connecsec-ted sec-t o each o sec-t h e r , b u sec-t sec-t h e y a r e k e p sec-t i n sec-t h e i r p o s i t i o n by means o f s t i f f s t r a i n - g a u g e dynamometers, w h i c h are con-n e c t e d t o a l o con-n g i t u d i con-n a l s t e e l box g i r d e r above the m.odel.

Table 1 . Main p a r t i c u l a r s o f t h e s h i p m o d e l . L e n g t h betv/een p e r p e n d i c u l a r s 2 . 2 5 8 m L e n g t h on t h e v / a t e r l i n e 2 . 2 9 6 m B r e a d t h 0 . 3 2 2 m Draught 0 . 1 2 9 m Volume o f d i s p l a c e m e n t 0 . 0 6 5 7 m ^ B l o c k c o e f f i c i e n t 0 . 7 0 0 C o e f f i c i e n t o f mid l e n g t h s e c t i o n 0 . 9 8 6 P r i s m i a t i c c o e f f i c i e n t 0 . 7 1 0 tVaterplane a r e a 0 . 5 7 2 m^ Waterplane c o e f f i c i e n t _{0 . 7 8 5}

k

0 . 1 6 8 5 m L o n g i t u d i n a l moment o f i n e r t i a o f w a t e r p l a n e 0 . 7 8 5

k

0 . 1 6 8 5 m L.C.B. f o r w a r d o f L / 2 PP C e n t r e o f e f f o r t o f w a t e r p l a n e a f t e r LPP / 2 0 . 0 1 1 m 0 . 0 3 8 m

Froude number o f s e r v i c e speed

a f t e r L / 2 PP

0 . 2 0

The dynamometers a r e s e n s i t i v e o n l y f o r f o r c e s p e r p e n d i c u l a r t o t h e b a s e l i n e o f t h e model.

By means o f a ScotchYoke miechanismi a harmonic h e a v i n g o r p i t -c h i n g m o t i o n -can be g i v e n t o t h e -c o m b i n a t i o n o f t h e seven s e -c t i o n s , w h i c h form t h e s h i p m o d e l . The t o t a l f o r c e s on each s e c t i o n c o u l d be measured as a f u n c t i o n o f f r e q u e n c y and speed.

A non segmented model o f t h e same form was a l s o t e s t e d i n t h e same c o n d i t i o n s o f f r e q u e n c y and speed t o compare t h e f o r c e s on t h e whole model w i t h t h e sum>B o f t h e s e c t i o n r e s u l t s . A p o s s i b l e e f f e c t o f t h e gaps between t h e s e c t i o n s c o u l d be d e t e c t e d i n t h i s way. The arrangement o f t h e t e s t s w i t h t h e segmented model and w i t h t h e whole model i s g i v e n i n F i g u r e 2 .

The m e c h a n i c a l o s c i l l a t o r and t h e measuring system i s shown i n F i g u r e 3« I n p r i n c i p l e t h e measuring system i s s i m . i l a r t o t h e one d e s c r i b e d by Goodman [ I I ] : t h e measured f o r c e s i g n a l i s m u l t i

8

o f t h e inphase and q u a d r a t u r e components can be found w i t h o u t d i s t o r t i o n due t o v i b r a t i o n n o i s e . I n some d e t a i l s t h e e l e c t r o n i c c i r -c u i t d i f f e r s somewhat from t h e d e s -c r i p t i o n i n [11]. I n p a r t i -c u l a r synchro r e s o l v e r s a r e used i n s t e a d o f s i n e - c o s i n e p o t e n t i o m e t e r s , because t h e y a l l o w h i g h e r r o t a t i o n a l speeds. The a c c u r a c y o f t h e i n s t r u m e n t a t i o n p r o v e d t o be s a t i s f a c t o r y w h i c h i s i m p o r t a n t f o r t h e d e t e r m i n a t i o n o f t h e q u a d r a t u r e compo-n e compo-n t s , w h i c h a r e s m a l l i compo-n comparisocompo-n w i t h t h e icompo-n-phase compocompo-necompo-nts o f t h e measured f o r c e s .

Throughout t h e e x p e r i m e n t s o n l y f i r s t harmonics were d e t e r m i n e d . I t s h o u l d be n o t e d t h a t n o n - l i n e a r e f f e c t s may be i m p o r t a n t f o r t h e s e c t i o n s a t t h e bow and t h e s t e r n where t h e s h i p i s n o t w a l l - s i d e d . The f o r c e d o s c i l l a t i o n t e s t s were c a r r i e d o u t f o r f r e q u e n c i e s up t o

CJ = 14 r a d / s e c . and f o u r speeds o f advance were c o n s i d e r e d , namely: Fn = . 1 5 , . 2 0 , ,25 and .30. Below a f r e q u e n c y o f ,CJ = 3 t o 4 r a d / s e c , t h e e x p e r i m e n t a l r e s u l t s a r e i n f l u e n c e d by w a l l e f f e c t due t o r e -f l e c t e d waves g e n e r a t e d by t h e o s c i l l a t i n g model.

The m o t i o n a m p l i t u d e s o f t h e s h i p m o d e l covered a s u f f i c i e n t l y l a r g e range t o s t u d y t h e l i n e a r i t y o f t h e measured v a l u e s (heave r j h cm, p i t c h / V 4,6 d e g r e e s ) . An example o f t h e measured f o r c e s on s e c t i o n 2, when t h e c o m b i n a t i o n o f t h e seven s e c t i o n s p e r f o r m s a p i t c h i n g m o t i o n , i s g i v e n i n F i g u r e 4,

3 . P r e s e n t a t i o n o f t h e r e s u l t s .

3 , 1 , Whole model.

I t i s assumed t h a t t h e f o r c e F and t h e moment M a c t i n g on a f o r c e d h e a v i n g o r p i t c h i n g s h i p m o d e l can be d e s c r i b e d by t h e f o l i o w i n g e q u a t i o n s : Heave; ( a + / 0 7 ) z + b i + cz = F sin(COt + «t) ' o o o z Dz +Ez +Gz = - M s i n ( c J t + (3) o o o z P i t c h ; ( A + k ^ ^C)ö + BO + 0 0 = M s i n ( Q t + ^ ) dS + eÓ + gO = - F ^ s i n C w t + S ) ( 1 ) ( 2 ) For a g i v e n h e a v i n g m o t i o n : z = z s i n t j t , i t f o l l o w s t h a t O cL b = a =

F since

z z 03 cL cz - F cos«t a z E = D = -^z ^^""/^ Gz + M cos /3 ' a z z r o 2 a'^ ( 3 ) S i m i l a r e x p r e s s i o n s a r e v a l i d f o r t h e p i t c h i n g m o t i o n .

The d e t e r m i n a t i o n o f t h e damping c o e f f i c i e n t s b and B and t h e dam-p i n g c r o s s - c o u dam-p l i n g c o e f f i c i e n t s e and E i s s t r a i g h t f o r w a r d : f o r a g i v e n f r e q u e n c y these c o e f f i c i e n t s a r e p r o p o r t i o n a l t o t h e qua-d r a t u r e components o f t h e f o r c e s o r moments f o r u n i t a m p l i t u qua-d e o f m o t i o n . For t h e d e t e r m i n a t i o n o f t h e added mass, t h e added mass moment o f i n e r t i a , a and A, and t h e added mass c r o s s - c o u p l i n g c o e f

f i c i e n t s d and D i t i s n e c e s s a r y t o know t h e r e s t o r i n g f o r c e and moment c o e f f i c i e n t s c and 0 , and t h e s t a t i c a l c r o s s c o u p l i n g c o e f -f i c i e n t s g and G.

The s t a t i c a l c o e f f i c i e n t s can be d e t e r m i n e d by e x p e r i m e n t s as a f u n c t i o n o f speed a t z e r o f r e q u e n c y . For heave t h e e x p e r i m e n

l o

-i n t h e a n a l y s -i s o f t h e t e s t r e s u l t s .

I n t h e case o f p i t c h i n g t h e r e i s a c o n s i d e r a b l e speed e f f e c t on the r e s t o r i n g moment c o e f f i c i e n t 0 . 0 decreases a p p r o x i m a t e l y 12?'o

when t h e speed i n c r e a s e s from Fn = . 1 5 t o . 3 0 . T h i s r e d u c t i o n i s due t o a hydrodynamic l i f t on t h e h u l l wlien t h e s h i p m o d e l i s towed w i t h a c o n s t a n t p i t c h a n g l e . O b v i o u s l y t h i s l i f t e f f e c t a l s o depends on the f r e q u e n c y o f t h e m o t i o n . C o n s e q u e n t l y , t h e c o e f f i c i e n t o f t h e r e s t o r i n g moment, as d e t e r m i n e d by an e x p e r i m e n t a t z e r o f r e q u e n c y , may d i f f e r from t h e v a l u e a t a g i v e n f r e q u e n c y .

As i t i s n o t p o s s i b l e t o measure t h e r e s t o r i n g moment and t h e s t a t i c a l c r o s s - c o u p l i n g as a f u n c t i o n o f f r e q u e n c y , i t was d e c i d e d t o use t h e c a l c u l a t e d v a l u e s a t z e r o speed. T h i s i s an a r b i t r a r y c h o i s e , w h i c h a f f e c t s t h e c o e f f i c i e n t s o f t h e a c c e l e r a t i o n t e r m s : f o r harmonic m o t i o n s a decrease o f C by AC r e s u l t s i n an i n c r e a s e of,A by ^ when C i s used i n t h e c a l c u l a t i o n .

. The r e s u l t s f o r t h e whole model a r e g i v e n i n t h e F i g u r e s 5 and 6, The r e s u l t s f o r t h e h e a v i n g m o t i o n were a l r e a d y p u b l i s h e d i n[ l 3 ] ; t h e y a r e p r e s e n t e d here f o r c o m p l e t e n e s s .

3 . 2 . P e s u l t s f o r t h e s e c t i o n s .

The components o f t h e f o r c e s on each o f t h e seven s e c t i o n s were d e t e r m i n e d i n t h e same way as f o r t h e whole model. As o n l y t h e f o r c e s and no moments on t h e s e c t i o n s were measured two equa-t i o n s remain f o r each s e c equa-t i o n :

Heave: ( a * +fiQ*')z + b * z + c*z = F* s i n ( 031 + ot* ) ' ' o o o z P i t c h : ( d * + ^ V * x ^ ) Ö + e 6 + gO = - F * s i n ( o 3 t +S*) (4) where V* i s t h e mass-moment o f t h e s e c t i o n i w i t h r e s p e c t t o t h e p i t c h i n g a x i s . The s t a r (*) i n d i c a t e s t h e c o e f f i c i e n t s o f t h e s e c t i o n s . Tho s e c t i o n c o e f f i c i e n t s d i v i d e d by t h e l e n g t h o f t h e s e c t i o n s g i v e t h e mean c r o s s - s e c t i o n c o e f f i c i e n t s , t h u s : a" L ~ 7 7 PP = a

and so on. Assuming t h a t t h e d i s t r i b u t i o n s o f t h e c r o s s - s e c t i o n a l v a l u e s o f t h e c o e f f i c i e n t s : a', b* e t c e t e r a , a r e c o n t i n u o u s c u r v e s , t h e s e d i s t r i b u t i o n s can be determiined from t h e seven mean c r o s s

-s e c t i o n v a l u e -s . I n t h e F i g u r e -s 7, 8, 9 and 10 t h e d i s t r i b u t i o n s o f t h e added mass a, t h e damjJing c o e f f i c i e n t b and t h e c r o s s c o u p l i n g c o e f f i c i e n t s d and e a r e g i v e n as a f u n c t i o n o f speed and f r e -quency. N u m e r i c a l v a l u e s o f t h e s e c t i o n r e s u l t s , a*, b* e t c e t e r a , a r e summarized i n the T a b l e s 2, 5 , 4 and 5.

I n F i g u r e 8 i t i s shown t h a t t h e d i s t r i b u t i o n o f t h e damping c o e f f i c i e n t b depends on f o r w a r d speed and f r e q u e n c y o f o s c i l l a t i o n . The damping c o e f f i c i e n t o f t h e f o r w a r d p a r t o f t h e s h i p m o d e l i n -c r e a s e s when t h e speed i s i n -c r e a s i n g . A t t h e same time a de-crease o f t h e damping c o e f f i c i e n t o f t h e a f t e r b o d y i s n o t i c e d . For h i g h f r e q u e n c i e s n e g a t i v e v a l u e s f o r t h e c r o s s s e c t i o n a l damping c o e f f i -c i e n t s are f o u n d .

The added mass d i s t r i b u t i o n , as shown i n F i g u r e 7t changes v e r y l i t t l e w i t h f o r w a r d speed b u t t h e r e i s a s h i f t f o r w a r d o f t h e d i s t r i b u t i o n curve f o r i n c r e a s i n g f r e q u e n c i e s .

N e g a t i v e v a l u e s f o r t h e c r o s s - s e c t i o n a l added ma.ss a r e found f o r t h e b o w - s e c t i o n s a t low f r e q u e n c i e s . For h i g h e r f r e q u e n c i e s t h e i n f l u e n c e o f f r e q u e n c y becomes v e r y s m a l l .

12

-The d i s t r i b u t i o n o f t h e damping c r o s s - c o u p l i n g c o e f f i c i e n t e v a r i e s w i t h speed and f r e q u e n c y as shown i n F i g u r e 10, From F i g u r e 9 i t can be seen t h a t t h e added mass c V o s s - c o u p l i n g c o e f f i c i e n t depends v e r y l i t t l e on speed. For h i g h e r f r e q u e n c i e s t h e i n f l u e n c e o f f r e q u e n c y i s s m a l l .

As a check on t h e a c c u r a c y o f t h e measurements t h e sum o f t h e r e s u l t s f o r the s e c t i o n s were compared w i t h t h e r e s u l t s f o r t h e whole model. The f o l l o w i n g r e l a t i o n s v;ere a n a l y s e d :

H a * = a

ƒ

d« X dx = A L C b * = b / e ' x d x = B L 5 I d * = d / a ' x d x = D L ETe* = e / ' b ' x d x = E

The r e s u l t s are shown i n F i g u r e 11 f o r a Froude number Fn = .20. For t h e o t h e r speeds a s i m i l a r r e s u l t was f o u n d . A n u m e r i c a l compa-r i s o n i s g i v e n i n the T a b l e s 2, 3, ^ and 5 . I t may be c o n c l u d e d t h a t t h e s e c t i o n r e s u l t s a r e i n agreement w i t h t h e v a l u e s f o r t h e whole model. No I n f l u e n c e o f t h e gaps between t h e s e c t i o n s c o u l d be f o u n d .

'f. A n a l y s i s o f t h e r e s u l t s .

The e x p e r i m e n t a l v a l u e s f o r the hydrodynamic f o r c e s and moments on the o s c i l l a t i n g s h i p m o d e l w i l l now be a n a l y s e d by u s i n g the s t r i p t h e o r y , t a k i n g i n t o a c c o u n t the e f f e c t o f f o r w a r d speed. For a de-t a i l e d d e s c r i p de-t i o n o f de-t h e s de-t r i p de-t h e o r y de-t h e r e a d e r i s r e f e r r e d de-t o [ l ] , [2] and [3J. For convenience a s h o r t d e s c r i p t i o n o f the s t r i p t h e o r y i s g i v e n h e r e . T h e ' t h e o r e t i c a l e s t i m a t i o n o f t h e hydrodynamic f o r c e s on a c r o s s s e c t i o n o f u n i t l e n g t h i s o f p a r t i c u l a r i n t e r e s t w i t h r e -g a r d t o t h e measured ' d i s t r i b u t i o n s o f t h e v a r i o u s c o e f f i c i e n t s a l o n -g t h e l e n g t h o f the s h i p m o d e l .

4.1, s t r i p t h e o r y ,

A r i g h t hand c o o r d i n a t e system x ^ y ^ z ^ i s f i x e d i n space. The

z ^ - a x i s i s v e r t i c a l l y upwards, the x ^ - a x i s i s i n t h e d i r e c t i o n o f t h e f o r w a r d speed o f t h e v e s s e l and t h e o r i g i n l i e s i n t h e u n d i s t u r b e d w a t e r s u r f a c e . A second r i g h t hand system o f a x i s x y z i s f i x e d t o t h e s h i p . The o r i g i n i s i n t h e c e n t r e o f g r a v i t y . I n t h e mean p o s i -t i o n o f -t h e s h i p -t h e body a x i s have -t h e same d i r e c -t i o n s as -t h e f i x e d a x i s ,

Consider f i r s t a s h i p p e r f o r m i n g a pure harmonic h e a v i n g mo-t i o n o f s m a l l a m p l i mo-t u d e i n s mo-t i l l w a mo-t e r . The s h i p i s p i e r c i n g a mo-t h i n s h e e t o f w a t e r , n o r m a l t o t h o f o r w a r d speed o f t h e s h i p , a t a f i x e d d i s t a n c e x f r o m t h e o r i g i n . At t h e t i m e t a s t r i p o f t h e s h i p a t a d i s t a n c e x from t h e c e n t r e o f g r a v i t y i s s i t u a t e d i n t h e s h e e t o f w a t e r . From x^ = V t + x i t f o l l o w s t h a t x = - V, where: V i s t h e speed o f t h e s h i p . The v e r t i c a l v e l o c i t y o f t h e s t r i p w i t h r e g a r d t o t h e w a t e r i s z^, t h e h e a v i n g v e l o c i t y . The o s c i l l a t o r y p a r t o f t h e h y d r o m e c h a n i -c a l f o r -c e on t h e s t r i p o f u n i t l e n g t h w i l l be:

where: ra' i s t h e added mass and N' i s t h e damping c o e f f i c i e n t f o r a s t r i p o f u n i t l e n g t h and y i s t h e h a l f w i d t h o f t h e s t r i p a t t h e w a t e r l i n e . Because: dm' _ dm * . dt ' "dT * ^ ' i t f o l l o w s t h a t : o 2 ƒ) g y z^, ( 5 ) For t h e whole s h i p we f i n d , ( 6 ) where A^ i s t h e w a t e r p l a n e a r e a .

14

-The moment produced by t h e f o r c e on t h e s t r i p i s g i v e n b y :

= -xFj» = ( x m ' ) Zo +(N« x - V x ~ - ) z ^ + 2 |» g x y z^ ( 7 ) Because J -x. —- dx = -m, we f i n d f o r t h e whole s h i p : L M„ = ( / x r a ' d x ) z + ( / N ' x d x + Vm)z + / j g S z ( 8 ) n £ ° L o / w o where i s t h e s t a t i c a l moment o f t h e w a t e r p l a n e a r e a . For a p i t c h i n g s h i p t h e v e r t i c a l speed o f t h e s t r i p a t x w i t h r e g a r d t o t h e w a t e r w i l l be: - x è + VO, and t h e a c c e l e r a t i o n i s : - x 8 + 2 V Ó . The v e r t i c a l f o r c e on t h e s t r i p w i l l be: F ' = - ~ m ' ( - x 6 + V 9 ) - N ' ( - x ö + V 0 ) + 2 ^ g y x 0 , d o r : = m' x 8 + ( N ' X - 2 Vm' - x V ^ ) 9 + ( 2; o g y x + ^ - N ' V) 0 ( 9 ) The t o t a l h y d r o m e c h a n i c a l f o r c e on t h e p i t c h i n g s h i p w i l l b e : F = ( / m ' x d x ) Ö + ( / l l ' x d x - V m ) Ó + ( og S - V/ n' d x ) © ( 1 0 ) ^ L L < L

The moment produced by t h e f o r c e on t h e s t r i p i s g i v e n b y : M' = - x F ' = . m ' x ^ ö - ( N ' x 2 - 2 Vm' x - x ^ V — • ) Ó -p -p dx - (2;»gy x^ + V ^ x ^ ,-N' V x ) e . ( 1 1 ) The t o t a l moment on t h e p i t c h i n g s h i p w i l l be M = - ( ƒ m'x^ dx)Ö - ( / N ' x ^ d x ) 0 - (/9g I - V^m - V / N ' x dx)ö, ( 1 2 ) T • T I W , P L because:-/ x ^ V ^ dx = - 2 v because:-/ m ' x d x . L L A summary o f t h e e x p r e s s i o n s f o r t h e v a r i o u s c o e f f i c i e n t s f o r t h e whole s h i p a c c o r d i n g t o t h e n o t a t i o n i n e q u a t i o r i s ( 1 ) and ( 2 ) i s g i v e n i n T a b l e 6.

Table 6. C o e f f i c i e n t s f o r t h e whole s h i p a c c o r d i n g t o t l i e s t r i p t h e o r y , a = ƒ m' dx d = / r a ' x d x + L L CJ2 b = / N ' d x e = / N ' x d x - V m L L c = ^ g g =

p

g S w (15) A = y r a X dx + —T' D = y m x d x L L B =

y

N' x ^ d x E

= ƒ

N' x d x + Vm L L w For t h e c r o s s - s e c t i o n a l v a l u e s o f t h e c o e f f i c i e n t s s i m i l a r ex p r e s s i o n s can be d e r i v e d from t h e e q u a t i o n s (5) t o ( 1 2 ) . For t h e comparison w i t h t h e e x p e r i m e n t a l r e s u l t s two o f these e x p r e s s i o n s a r e g i v e n h e r e , namely: b' . N' - V ^ dx ( 1 4 ) ' • • dm' e = N x - 2 V m - x V ^ dx A l s o i t f o l l o w s t h a t : A =

ƒ

d ' X dx and: (15) B s:

J

e X dx

16

-4 . 2 . Comparison o f t h e o r y and e x p e r i m e n t .

For a number o f cases t h e e x p e r i m e n t a l r e s u l t s a r e compared w i t h t h e o r y . F i r s t o f a l l t h e damping c r o s s - c o u p l i n g c o e f f i c i e n t t are c o n s i d e r e d . From e q u a t i o n s (13) i t f o l l o w s t h a t : E = / N ' x d x + Vm (16) X d x - V m The f i r s t term i n b o t h e x p r e s s i o n s i s t h e c r o s s c o u p l i n g c o e f f i -c i e n t f o r z e r o f o r w a r d speed. For a f o r and a f t s y m m e t r i -c a l s h i p t h i s term i s e q u a l t o z e r o . For such a s h i p t h e r e s u l t i n g e x p r e s -s i o n -s are e q u a l i n magnitude b u t have o p p o -s i t e -s i g n , w h i c h i -s i n agreement w i t h t h e r e s u l t found by Timman and Newman [?]. The e x p e r i ments c o n f i r m t h i s f a c t as shown i n F i g u r e 13 where e and E a r e p l o t t e d on a base o f f o r w a r d speed as a f u n c t i o n o f t h e f r e q u e n c y o f o s c i l l a t i o n . The magnitude o f t h e speed dependent p a r t s o f t h e c o e f f i -c i e n t s i s e q u a l w i t h i n v e r y -c l o s e l i m i t s . E x t r a p o l a t i o n t o z e r o speeds shows t h a t t h e e and E l i n e s i n t e r s e c t i n one p o i n t w h i c h s h o u l d r e p r e s e n t t h e z e r o speed c r o s s - c o u p l i n g c o e f f i c i e n t .

U s i n g Grim's t w o - d i m e n s i o n a l s o l u t i o n f o r damping and added mass a t z e r o speed [9]t h e c o e f f i c i e n t s e and E were a l s o c a l c u l a t e d a c c o r d i n g t o t h e e q u a t i o n s ( 1 6 ) . The d i s t r i b u t i o n o f added mass and damping c o e f f i c i e n t f o r z e r o speed i s g i v e n i n F i g u r e 12 and t h e c a l c u l a t e d damping c r o s s - c o u p l i n g c o e f f i c i e n t s a r e shown i n F i g u r e 13.

The c a l c u l a t e d v a l u e s a r e i n l i n e w i t h t h e e x p e r i m e n t a l r e s u l t s . The n a t u r a l f r e q u e n c i e s f o r p i t c h and heave a r e r e s p e c t i v e l y 0 = 7 . 0 / 6.9 rad/sec and i n t h i s i m p o r t a n t r e g i o n the c a l c u l a t i o n o f t h e dam-p i n g c r o s s - c o u dam-p l i n g c o e f f i c i e n t s i s q u i t e s a t i s f a c t o r y . The z e r o sdam-peed case w i l l be s t u d i e d i n t h e near f u t u r e by o s c i l l a t i n g e x p e r i m e n t s i n a wide b a s i n t o a v o i d w a l l i n f l u e n c e . A n o t h e r c o m p a r i s o n o f t h e o r y and e x p e r i m e n t c o n c e r n s t h e d i s t r i -b u t i o n a l o n g t h e l e n g t h o f the s h i p m o d e l o f t h e damping c o e f f i c i e n t and o f t h e damping c r o s s - c o u p l i n g c o e f f i c i e n t e.

From e q u a t i o n ( l 4 ) :

dx

e' = N' X - 2 Vm* - x V ^

dx

A g a i n u s i n g Grim's t w o - d i m e n s i o n a l v a l u e s f o r n' and m*, these d i s -t r i b u -t i o n s c o u l d be c a l c u l a -t e d . An example i s g i v e n i n F i g u r e l 4 . A l s o i n t h i s case t h e agreement between the c a l c u l a t i o n and t h e exp e r i m e n t i s good. For h i g h sexpeeds n e g a t i v e v a l u e s o f t h e c r o s s s e c -t i o n a l damping i n -t h e a f -t e r b o d y can be e x p l a i n e d on -t h e b a s i s o f t h e e x p r e s s i o n f o r b', because i n t h a t r e g i o n i s a p o s i t i v e quan-t i quan-t y .

F i n a l l y t h e v a l u e s f o r t h e c o e f f i c i e n t s A, B, a and b f o r t h e whole model, as g i v e n by t h e e q u a t i o n s (13) were c a l c u l a t e d and com-p a r e d w i t h t h e e x com-p e r i m e n t a l r e s u l t s . F i g u r e 15 shows t h a t t h e damcom-p- dampi n g dampi n p dampi t c h dampi s over e s t dampi m a t e d f o r low f r e q u e n c dampi e s . The o t h e r c o e f -f i c i e n t s agree q u i t e w e l l w i t h t h e e x p e r i m e n t a l r e s u l t s .

18

-TABLE 2,

Added mass f o r t h e s e c t i o n s and the whole model. kg sec /m. Fn = .15» CO _a * a r a d / s e c 1 . 2 3 4 ₅ 6 7 sura o f s e c t i o n s whole model -1 ,21- 0,59 — _{0,54 0,87}

0,41

_{-0,17 1}

,84

6 _0,31 0,66 1,08 1,38 1,26 0,65 0,02 5,36 5,37 8 0,24 o,6o 1,09 1,37 1,28 0,76 0,10 5,44 5,26 10 0,20 0,69 1,29

1,48

1,34 0,85 0,14 _{5,99 5,91} 12 0,18' 0,78 1,4o 1,60 1,45 0,90 0,17

6,48

6,39 Fn = .20. 0,59 0,83 1,29 1,59 1,15 0,22 -0,27 5,40 5,63 6 0,32 0,65 1 ,00 1,4o 1,23

0,64

0

5,24

5,19 • 8 0,21 ₀,55 1,08 1,38 1 ,21 0,75 0,12 5,30 5,18 10 0,19 0,65 1,23 1,49 1,33 0,83 o,i4 5,86 5,78 12 0,20 0,77 1,37 1,60 1,45 0,88 0,17 6,44 6,32 Fn : = .25. 0,86 1,09 1,26 1 ,66 1 ,20 0,16 -0,32 5,91 4,99 6 _0,33 0,65 1,01 1,38 1,19 0,55 -0,02 5,09 4,89 8 0,20 0,54 1,03 1,39 1 ,26 0,68 0,08 5,18 5,15 . 10 0,18 0,62 _1,19

1,48

1,34 0,77 0,12 5,70 5,65 12 0,20 0,76 1,37 1,60 1 ,45 0,83 0,16 6,37 6,21 • _{Fn =} = .30. 0,70 _0,91 1,49 1,58 1,07 -0,10 -0,22 5,43 5,59 • 6 0,25 0,44. ₁,15 1,39 1,07 0,45 0,07 ^,82 4,51 8 0,16

0,42

1,14 1,45 1,08 0,58 0,13 4,96 4,93

•10

0,15 0,55 1,26 1 ,47 1 ,22 0,68 0,17 5,50

5,48

12 0,17 0,69

1,41

1,57 1,35 0,81 0,19 6,19 6,18

Damping c o e f f i c i e n t s f o r t h e s e c t i o n s and t h e whole model. kg sec/m. Fn o .13. CO • b b r a d / sum o f whole sec 1 2 3 4 5 6 7 s e c t i o n s model 2,03 9,78 5,78 3,80 4,80 2,00

-

35,63 .. 6 1,82 4,42 4,55 4,58 4,52 4,78 1,67 26,34 26,53 8 1,61 _2,31 2,26 2,75 3,35 3,94 1,53 17,75 17,49 10 1,56 1,08 0,76 1,39 2,36 3.43 1,49 11,87 11 ,63 12 _0,95 0,47 0,44 0,87 1,89 3,09 1,50 9,21 8,54 Fn = .20. 4 1,33 4,53 5,08 5,05 5,73 6,63 2,50 31,05 31 ,35 6 1,95 3,95 4,32 4,45 4,52 5,07 . 2,07 26,33 26,15 , 8 ; 1,50 • 1,91 2,25 2,81 3,49 4,38 1,94 18,28 17,78 10 1,10 0,37 0,62 1,54 2,70 4,01 1,90 12,24 12,14 12 0,74 -0,15 0,21 1 ,01 2,18 3,84 _1,93 9,76 9,03 Fn = .25. 4 2,13 4,80 5,38 5,20 5,98 7,63 2,85 33,97 35,88 6 _{1,97 3,43 4,17 4,23 , 4,62} 5,68 §,35 26,45 27,63 8 1,48 1,58 2,28 2,83 3,68 5,21 _2,19 19,25 18,75 10 0,95 -0,06 0,60 1,68 3,00 4,96 2,20 13,33 12,69 12 0,52 -0,56 -0,03' 1,03 2,63 4,74 2,29 10,62 9,78 Fn = . 30. 4 1,78 4,40 4,40 5,15 6,78 7,60 2,98 33,09 38,10 6 1,75 2,77 3,50 4,10 5,18 6,32 2,55 26,17 28,45 8 1,21 0,99 1,70 2,81 4,50 5,73 2,51 19,45 20,4o 10 0,64 -0,87 0,17 1,88 4,07 5,42 2,59 13,90 13,95 12 0,42 -0,56 -0,63 1,37 3,72 5,28 2,66 11,26 10,4'

20

-TABLE k.

Added mass c r o s s - c o u p l i n g c o e f f i c i e n t s f o r t h e s e c t i o n s and t h e v/hole model.

2 kg sec . Fn = .15. CO

•

d d r a d / s e c 1 2 3 4 5 6 7 sum o f s e c t i o n s whole model k — _ —

-

_+0,59 +0,28

-

-

-6 . -0,42 -0,47 -0,33 +0,02 +0,46 +0,57 +0,13 -0,04 +0,09 8 -0,27 -0,44 -0,40 -0,01 +0,38 +0,50 +0,13 -0,11 -0,16 10 -0,19 -0,43 -0,4o -0,01 +0,37 +0,49 +0,15 -0,02 -0,10 12 -0,19 -0,45 -0,4o -0,01 +0,40 +0,51 +0,15 +0,01 -o,o4 • Fn = = .20. . ,4 -0,57 -0,67

-

-

-

+0,78 +0,32

-

-6 -0,39 -0,52 -0,34 +0,01 +0,46 +0.,59 +0,13 -0,06 -0,06 8 -0,24 -0,45 -0,40 -0,01 _+0,39 +0,51 +0,11 -0,09 -0,14 10 -0,20. -0,45 -0,4o -0,01 +0,38 +0,51 +0,13 -o,o4 -0,08 12 . -0,20 -0,47 -0,41 -0,01• +0,40 +0,53 +0,14 -0,02 -0,03 Fn = = .25. -0,62 -0.59 -0,01 +0,12 +0,72 +0,86 +0,21 +0,69 +0,15 6 -0,39 -0,50 -0,32 +0,02 +0,46 +0,59 +0,13 -0,01 0,00 • 8 -0,25 -0,48 -0,40 -0,01 +0,39 +0,52 +0,14 -0,07 -0,13 10 -0,18 -0,46 -0,42 -0,01 +0,38 +0,51 +0,13 -0,05 -0,08 12 -0,20 -0,46 -0,42 . -0,01 +0,40 +0,51 +0,15 -0,03 -0,05 Fn = = .30. . h -0,62 -0,61 +0,13 +0,08 +0,64 +0,95 +0,20 +0,75 +1,09 6 -0,29 -0,47 -0,36 +0,01 +0,43 +0,59 +0,21 +0,12 +0,01 8 -0,21 -0,47 -0,44 -0,01 +0,38 +0,53 +0,16 -0,06 -0,11 • 10 -0,19 -0,46 -0,44 -0,02 +0,38 +0,51 +0,15 -0,07 -0,10 12 -0,20 -0,46 -0,44 -0,02 +0,39 +0,52 +0,16 -0,05 -0,06

Damping c r o s s - c o u p l i n p ; c o e f f i c i e n t s f o r t h e s e c t i o n s and t h e whole model»

kg s e c . CO <« e e r a d / sec 1 2 3 4 5 6 7 sum o f s e c t i o n s whole model k

-

-

-

-

+1,63 +1,34

-

-

- 2,43 6 -1,65 -2,58 -2,12 -1,19 -0,09 +1,70 +1 ,21 - 4,72 - 5,32 8 -1,71 -2,49 -2,45 -1,81 -0,68 +1,20 +1,09 - 6,84 - 6,75 10 -1,4o -2,01 -2,43 -2,10 -1,21 . +0,88 +1,05 - 7,22 - 7,04 12 -1,07 -1,55 -2,28 -2,39 -1,52 +0,63 +1 ,05 - 7.13 - 6,88 Fn : = .20, k -1 ,22 -3,07

-

-

-

+2,39 +1,77

-

- 6,63 6. -1,68 -2,43 -2,40 -2,06 -0,68 +1,52 +1,42 - 6,31 - 6,65 8 -1,59 -2,36 -2,83 -2,50 -1,25 +1,11 +1,32 - 8,10 - 8,23 10 -1,29 -2,04 -3,02 -2,87. -1,75 +0,82 +1,29 - 8,86 - 8,86 12 -0,98 -1,65 -2,99 -2,97 -2,06 +0,61 +1,30 - 8,74 - 8,75 Fn : = .25. k -1,52 -3,04 -5,47 -3,03 -0,96 +2,16 +1,91 - 7,95 - 6,70 6 -1,50 -2,21 -2,83 -2,66 -1,36 +1,47 +1,61 = 7,50 - 7,38 8 -1,50 -2,26 -3,21 -2,97 -1,79 +1,11 +1,51 - 9,11 - 9,30 10 -1,22 -2,14 -3,56 -3,39 -2,27 +0,86 +1,49 -10,23 -10,18 12 -0,85 -1,81 -3,66 • -3,58 -2,53 +0,66 +1,47 -10,30 -10,31 • Fn : = ,30, k -1,37 -2,82 -3,61 -3,06 -1,22 +2,19 +1,98 -.7,91 - 7,55 6 -1,23 -1,93 -3,16 -3,06 -1,84 +1.43 +1,72 - 8,07 - 7,95 8 -1,30 -1,96 -3,55 -3,42 -2,32 +i-,03 +1,67 - 9,85 - 9,81 10 _{-1,19 -2,06} -3,94 -3,90 -2,70 +0,76 +1,67 -11,36 -11,25 12 -0,91 -1,97 -4,08 -4,19 • -2,97 +0,56 +1,69 -11,87 -11,84

L i s t o f symbols. a . • g A . , G a*. . g* A*. . G* a' . . g' A''. . G' Fn s k yy L PP m' N' t Y x y z X ,y ,z © C o e f f i c i e n t o f t h e m o t i o n e q u a t i o n s ( h y d r o m e c h a n i c a l p a r t ) . The same f o r a s e c t i o n o f t h e s h i p . The same f o r a c r o s s - s e c t i o n o f t h e s h i p . B l o c k c o e f f i c i e n t . Froude number. A m p l i t u d e o f v e r t i c a l f o r c e on a h e a v i n g o r p i t c h i n g s h i p . O s c i l l a t o r y p a r t o f t h e h y d r o m e c h a n i c a l f o r c e on a h e a v i n g o r p i t c h i n g s h i p . A c c e l e r a t i o n o f g r a v i t y . L o n g i t u d i n a l r a d i u s o f i n e r t i a o f t h e s h i p . L e n g t h between p e r p e n d i c u l a r s . A m p l i t u d e o f moment on a h e a v i n g o r p i t c h i n g s h i p . O s c i l l a t o r y p a r t o f t h e h y d r o m e c h a n i c a l moment on a h e a v i n g o r p i t c h i n g s h i p . Added mass o f a c r o s s - s e c t i o n ( z e r o s p e e d ) . Damping c o e f f i c i e n t o f a c r o s s - s e c t i o n ( z e r o s p e e d ) . Time. . Forward speed o f s h i p . R i g h t hand c o o r d i n a t e system, f i x e d t o t h e s h i p . R i g h t hand c o o r d i n a t e system, f i x e d i n space. V e r t i c a l d i s p l a c e m e n t o f s h i p .

D i s t a n c e o f c e n t r e o f g r a v i t y o f a s e c t i o n t o t h e p i t c h i n g a x i s .

Phase a n g l e s . P i t c h a n g l e .

D e n s i t y o f w a t e r . C i r c u l a r f r e q u e n c y .

Volume o f d i s p l a c e m e n t o f s h i p . Volume o f d i s p l a c e m e n t o f s e c t i o n .

2h -5. R e f e r e n c e s . 1 . B.V. K o r v i n - K r o u k o v s k y , V/.R. Jacobs. " P i t c h i n g and h e a v i n g m o t i o n s o f a s h i p i n r e g u l a r waves". S.N.A.M.E. 1957. 2, J.A. Fay.

"The m o t i o n s and i n t e r n a l r e a c t i o n s o f a v e s s e l i n r e g u l a r waves". J o u r n a l o f S h i p Research 1958.

3» Y. Watanabe.

"On t h e t h e o r y o f p i t c h and heave o f a s h i p " .

Technology Reports o f t h e Kyushu U n i v e r s i t y . V o l . 31 No. 1, 1958. E n g l i s h t r a n s l a t i o n by Y. Sonoda, 1963»

h, J . Fukuda.

"Coupled m o t i o n s and m i d s h i p b e n d i n g moments o f a s h i p i n r e g u l a r waves".

. ^ J o u r n a l o f t h e S o c i e t y o f Naval A r c h i t e c t s o f Japan, No. 112, 19^2. 5. J . G e r r i t s m a . " S h i p m o t i o n s i n l o n g i t u d i n a l waves". I n t e r n a t i o n a l S h i p b u i l d i n g P r o g r e s s I 9 6 O . 6. G. Vossers. "Some a p p l i c a t i o n s o f t h e s l e n d e r body t h e o r y i n s h i p h y d r o d y n a m i c s " . T h e s i s D e l f t I 9 6 2 , 7. R. Timman, J.N. Newman,

"The c o u p l e d damping c o e f f i c i e n t o f a symmetric s h i p " . • J o u r n a l o f S h i p Research, I 9 6 2 .

8. P. G o l o v a t o .

"The f o r c e s and moments on a h e a v i n g s u r f a c e s h i p " . J o u r n a l o f S h i p Research 1957.

9. 0. Grim.

"A method f o r a more p r e c i s e c o m p u t a t i o n o f h e a v i n g and p i t c h i n g m o t i o n s b o t h i n sm.ooth w a t e r and i n waves".

10, F. T a s a i .

a, "On t h e damping f o r c e and added mass o f s h i p s h e a v i n g and p i t -c h i n g " .

b, "Measurements o f the- w a v e h e i g h t produced by t h o f o r c e d h e a v i n g of t h e c y l i n d e r s " ,

c, "On t h e f r e e h e a v i n g o f a c y l i n d e r f l o a t i n g on t h e s u r f a c e o f a f l u i d " .

R e p o r t s o f Research I n s t i t u t e f o r A p p l i e d Mechanics. Kyushu U n i v e r s i t y , Japan. V o l . V I I I I96O,

11, A. Goodman.

" E x p e r i m e n t a l t e c h n i q u e s and methods o f a n a l y s i s used i n submerged b o d y research''.

T h i r d Symposium on N a v a l Hydrodynamics, Scheveningen I96O. 12. H.J. Z u n d e r d o r p , M. B u i t e n h e k .

" O s c i l l a t o r t e c h n i q u e s a t t h e S h i p b u i l d i n g L a b o r a t o r y " .

Report no. 111. S h i p b u i l d i n g L a b o r a t o r y , T e c h n o l o g i c a l U n i v e r s i t y , D e l f t , 1963,

13, J . G e r r i t s m a , W. Beukelman,

" D i s t r i b u t i o n o f dam.ping and added mass a l o n g t h e l e n g t h o f a s h i p

-m o d e l " , .

HEAVE AFTER PITCH

0.75 1.00 1.25 1.50 1.75

X / i ^ W A V E LENGTH RATIO

I N F L U E N C E OF C R O S S ^ C O U P L I N G

(W&S 5 3 2 0 )

F I G U R E 1

I

£ F * sin(ü)t + a * )

i6i

1 2 3 4 5 6 7 HEAVING TEST WITH SEGMENTED MODEL

F j * s i n ( a ) t + 6 j * )

1 2 3 4 5 6 PITCHING TEST WITH SEGMENTED MODEL

0) ca W W

W t

HEAVING TEST WITH WHOLE MODEL PITCHING TEST WITH WHOLE MODEL

ARRANGEMENT OF OSCILLATION TESTS

FIGURE 2

(W&S 5 3 2 0 )

ELECTRONIC STRAIN INDICATOR CARRIER AMPLIFIER _RESOLVER

AMPLIFIER

DEMODULATOR

T = INTEGRATOR

S T R A I N GAUGE DYNAMOMETER IN PHASE COMPONENT QUADRATURE COMPONENT

PRINCIPLE OF MECHANICAL OSCILLATOR AND ELECTRONIC CIRCUIT

F n =

.20

Q U A D R A T U R E COMPONENT

1 2

CIRCULAR FREQUENCY ^adƒsec IN P H A S E COMPONENT ^ 0 - L • r =

1

c m • r =

2

c m A r =

3

c m I I I I ' I 1 ! L

COMPONENTS OF FORCE ON SECTION 2 . PITCHING MOTION

FIGURE A

HEAVING MOTION o 4 * 'i 10 (JÜ. 15 Fn =.15 Fn = .20 F n =.25 F n =.30

EXPERIMENTAL RESULTS FOR WHOLE MODEL

0)

10

rad/sec

15 0) 10

rad/sec

15 U w O) 2', Ul y t UJ O u O z 0 . O O _1 _2 3 h -0) 10 rad/sec 15 10 15 CO Fn = .15 _ _ F n = .20 F n =.25 _.._Fn = .30

rad/; sec

EXPERIMENTAL RESULTS FOR WHOLE MODEL

F n . , 1 5 F n . . 2 0

F n r . 2 5 F n = .30

DISTRIBUTION OF d'' OVER THE LENGTH OF THE SHIPMODEL

'•• r o d / . ï ' 1 ^ -(fl A . (H - 2" - 2" i i 10 1 • • I I I ' I

!

i ] 1 i 5 10 • S U M OF S e C I I O M S O W H O L E M O D E L S »

C O M P A R I S O N OF THE S U M S OF SECTION R E S U L T S AND T H E WHOLE MODEL R E S U L T S FOR F R O U D E NUMBER F n = . 2 0

F I G U R E 11

DISTRIBUTION O F e OVER T H E L E N G T H O F THE SHIPMODEL (W&S 5 3 2 0 ) _{F I G U R E 10}

( i ) _ » r o d / s 10 15 Fn..15 F11..JO Fn..J5 ll) ^ rad/s 10 15 Fn ..30

COMPARISON OF CALCULATED AND MEASURED VALUES FOR a . b . A AND B ( W H O L E MODEL) F I G U R E 15

DISTRIBUTION O F a OVER T H E L E N G T H OF T H E S H I P M O D E L (WtS 5 3 2 0 ) F I G U R E 7